In previous courses, students worked with percent change and representing percent change with an exponential function. They learned about the relationship between a growth rate and growth factor The purpose of this Warm-up is to prepare students to recognize 20% growth and use a growth factor of 1.2 to represent it when it appears later in this lesson.

The goal of this discussion is to check that students understand the difference between growth rate and growth factor when talking about a sequence. Begin by selecting students to share how they calculated the growth factor and why the number they chose for the percent makes sense.

Depending on what students recall of growth factor from previous courses, consider highlighting:

• 1.5 is known as the growth factor for this function, and 0.5 is known as the growth rate.
• Growth rate is often expressed as a percentage, so 50%.
• The result of increasing a quantity by 50% can be calculated by multiplying it by 1.5.
• More generally, the result of increasing a quantity by % can be calculated by multiplying it by .

It is not required that students memorize the vocabulary “growth rate” and “growth factor” at this time, but they should recognize that both are ways to describe how the value of a sequence changes from term to term.

If time allows, conclude the discussion by asking students:

• “How can you define the term of the sequence?” ( for )
• “Where in that function do you see the growth rate and growth factor?” (The growth factor is the value being repeatedly multiplied for different values of . The growth rate uses a percent to describe how much more (or less) the value of one term is when compared to the previous term, so in this case it is the growth factor minus 1, which is 0.5.)

If students need additional practice differentiating between growth rate and growth factor , display the following questions for all to see and ask students to calculate the growth factor and growth rate for each.

1. 64, 80, 100, 125 (1.25 and 25%)
2. 64, 112, 196, 343 (1.75 and 75%)
3. 64, 128, 256, 512 (2 and 100%)
4. 125, 100, 80, 64 (0.8 and 20% (decrease))

After a brief work time, select students to share how they calculated their solutions.

Math Community
After the Warm-up, display the revisions to the class Math Community Chart that were made from student suggestions in an earlier exercise. Tell students that over the next few exercises, this chart will help the class decide on community norms—how they as a class hope to work and interact together over the year. To get ready for making those decisions, students are invited at the end of today’s lesson to share which “Doing Math” action on the chart is most important to them personally.

This is the first of two activities in which students define sequences with equations and use their equations to answer questions about the context (MP2). The population values were purposefully chosen in order for students to focus on creating representations (like tables and graphs) and not on calculating “best fit.”

Making spreadsheet and graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

This is the second of two activities in which students define sequences with equations and use their equations to answer questions about the context. Similar to the previous activity, students are asked to "write an equation" without being told the type, so the choice is theirs to define the sequences recursively or for the term, after they make sense of the problem (MP1). Making spreadsheet and graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Monitor for students who use different strategies to identify the type of sequences represented by the black and white squares. Here are some examples of strategies students may use, ordered from concrete to abstract:

• Graphing the number of white or black squares for each step
• Making a table for the total number of white or black squares for each step and then calculating the differences between successive terms
• Visualizing ways that the white squares grow to write equations in order to tell that is arithmetic
• Visualizing ways that the black squares grow to write equations in order to tell that is neither arithmetic nor geometric

If a strategy is not used, it is not necessary to introduce it.

The routine of Anticipate, Monitor, Select, Sequence, Connect (5 Practices) requires a balance of planning and flexibility. The anticipated approaches might not surface in every class, and there may be reason to change the order in which strategies are presented. While monitoring, keep in mind the learning goal and adjust the order to ensure all students have access to the first idea presented (whether that be a common misconception or a different approach).